Blockchain (BC) systems are highly distributed peer-to-peer networks that offer an alternative to centralized services and promise robustness to coordinated attacks. However, the resilience and overall security of a BC system rests heavily on the structural properties of its underlying peer-to-peer overlay. Despite their success, BC overlay networks' critical design aspects, connectivity properties and network-layer inter-dependencies are still poorly understood. In this work, we set out to fill this gap and study the most important overlay network structural properties and robustness to targeted attacks of seven distinct BC networks. In particular, we probe and crawl these BC networks every two hours to gather information about all their available peers, over a duration of 28 days. We analyze 335 network snapshots per BC network, for a total of 2345 snapshots. We construct, at frequent intervals, connectivity graphs for each BC network, consisting of all potential connections between peers. We analyze the structural graph properties of these networks and compare them across the seven BC networks. We also study how these properties associate with the resilience of each network to partitioning attacks, i.e., when peers are selected, attacked and taken offline, using different selection strategies driven by the aforementioned structural properties. In fact, we show that by targeting fewer than 10 highly-connected peers, major BCs such as Bitcoin can be partitioned into disjoint, i.e., disconnected, components. Finally, we uncover a hidden interconnection between different BC networks, where certain peers participate in more than one BC network, which has serious implications for the robustness of the overall BC network ecosystem.